Determine how many solutions exist for the system of equations. ${12x-2y = 14}$ ${y = 10+5x}$
Solution: Convert both equations to slope-intercept form: ${12x-2y = 14}$ $12x{-12x} - 2y = 14{-12x}$ $-2y = 14-12x$ $y = -7+6x$ ${y = 6x-7}$ ${y = 10+5x}$ ${y = 5x+10}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = 6x-7}$ ${y = 5x+10}$ The linear equations have different slopes. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ When two equations have different slopes, the lines will intersect once with one solution.